In this paper, we discuss how access to health-related data by private insurers, other than affecting the interests of prospective policy-holders, can also influence their propensity to make personal data available for research purposes. We take the case of national precision medicine initiatives as an illustrative example of this possible tendency. Precision medicine pools together unprecedented amounts of genetic as well as phenotypic data. The possibility that private insurers could claim access to such rapidly accumulating biomedical Big Data or to health-related information derived from it would discourage people from enrolling in precision medicine studies. Should that be the case, the economic value of personal data for the insurance industry would end up affecting the public value of data as a scientific resource. In what follows we articulate three principles – trustworthiness, openness and evidence – to address this problem and tame its potentially harmful effects on the development of precision medicine and, more generally, on the advancement of medical science.